Abstract

Data from photomultiplier tubes are typically analyzed using either counting or averaging techniques, which are most accurate in the dim and bright signal limits, respectively. A statistical means of adjoining these two techniques is presented by recovering the Poisson parameter from averaged data and relating it to the statistics of binomial counting from Kissick et al. [Anal. Chem. 82, 10129 (2010)]. The point at which binomial photon counting and averaging have equal signal to noise ratios is derived. Adjoining these two techniques generates signal to noise ratios at 87% to approaching 100% of theoretical maximum across the full dynamic range of the photomultiplier tube used. The technique is demonstrated in a second harmonic generation microscope.

Figures (3)

(Left) SNR of photon averaging and binary counting. (Right) SNR of photon averaging and binary counting as a ratio of the theoretical maximum SNR (the SNR of the underlying Poisson distribution). SNR at the crossover point is at ~87% of the theoretical limit. μ1 = 7.2 mV, σ1 = 4 mV, σJ = 0.3 mV. Data were simulated per sample by summing a Poisson distributed random number of lognormal random numbers, with an additional normally distributed random number also added to represent the Johnson noise (between 2 × 106 values at low λ to 5000 at high λ). Binary counting offers higher SNR at low λ, and photon counting offers higher SNR at high λ.

SHG images of crystalline urea. (Left Column) Full contrast image, (Right Column) Contrast adjusted to λmax = 0.02. (A,B) Analysis with photon averaging only. The majority of the image is silhouetted in comparison to the brightest pixels in (A), up to λ = 74. The dimmest pixels are evident in (B), but are as prominent as the horizontal streaks and noise in the image. (C,D) Analysis with binomial counting only. The largest recoverable value was λ = 6.23; pixels brighter than this were clipped to this value in (C). The dimmest pixels are easily identifiable in (D), and the instrument noise is not evident. (E,F) Preferential crossover analysis incorporating photon averaging and binomial counting. Pixels brighter than λ = 0.48 were analyzed by photon averaging, so the full upper range of detection is preserved in (E). Pixels dimmer than λ = 0.48 were analyzed by binomial counting, so the lower range of detection is preserved in (F). Selection of a crossover point defined by Eq. (18) maximizes SNR across the entire range of detection.